Asymptotic Analysis and Singular Perturbation Theory

Energy eigenstates are wavefunctions of the form
ψ(t) = e −iEt/ ϕ,
where ϕ ∈ H and E ∈ R. It follows from the Schrödinger equation that
Hϕ = Eϕ.
Hence E is an eigenvalue of H and ϕ is an eigenvector. One of Schrödinger’s
motivations for introducing his equation was that eigenvalue problems led to the
experimentally observed discrete energy levels of atoms.
Now suppose that the Hamiltonian
H ε = H0 + εH1 + O(ε 2 )
depends smoothly on a parameter ε. Then, rewriting the previous result, we find
that the corresponding simple energy eigenvalues (assuming they exist) have the
expansion
E ε = E0 + ε 〈ϕ0, H1ϕ0〉
〈ϕ0, ϕ0〉 + O(ε2 )
where ϕ0 is an eigenvector of H0.
For example, the Schrödinger equation that describes a particle of mass m moving
in R d under the influence of a conservative force field with potential V : R d → R
is
iψt = − 2
∆ψ + V ψ.
2m
Here, the wavefunction ψ(x, t) is a function of a space variable x ∈ R d and time
t ∈ R. At fixed time t, we have ψ(·, t) ∈ L 2 (R d ), where
L 2 (R d ) = u : R d → C | u is measurable and
R d |u| 2 dx < ∞
is the Hilbert space of square-integrable functions with inner-product
〈u, v〉 = u(x)v(x) dx.
R d
The Hamiltonian operator H : D(H) ⊂ H → H, with domain D(H), is given by
H = − 2
∆ + V.
2m
If u, v are smooth functions that decay sufficiently rapidly at infinity, then Green’s
theorem implies that
〈u, Hv〉 =
Rd
u − 2
∆v + V v dx
2m
2 2
=
∇ · (v∇u − u∇v) − (∆u)v + V uv dx
2m 2m
R d
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